## Find the orthogonal projection

1. The problem statement, all variables and given/known data

My questions is this:
How to find the orthogonal projection of vector y= (7,-4,-1,2) on null space
N(A)

Where A is a matrix
A =

$$\left(\begin{array}{cccc}2&1&1&3\\3&2&2&1\\1&2&2&-9\end{array}\right)$$

2. Relevant equations

$$A^TA\overline{x}=A^T\overline{y}$$

3. The attempt at a solution
First I found the Null space of matrix A:
A =

$$\left(\begin{array}{cc}0&-5\\-1&7\\1&0\\0&1\end{array}\right)$$

Then, I applied he formula from aboce:

A^TA =
2 -7
-7 75

A^Ty= (3,-61)

after that built an equation to find x:

$$\left(\begin{array}{cc}2&-7\\-7&75\end{array}\right) \left(\begin{array}{c}X1\\X2\end{array}\right) = \left(\begin{array}{c}3\\-61\end{array}\right)$$

x1 = -2 , x2=-1
P(x) = (5,-5,-2,-1)